Related papers: Proof of two supercongruences of truncated hyperge…

By examining asymptotic behavior of certain infinite basic ($q$-) hypergeometric sums at roots of unity (that is, at a "$q$-microscopic" level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial…

In 2017, motivated by a supercongruence conjectured by Kimoto and Wakayama and confirmed by Long, Osburn and Swisher, Z.-W. Sun introduced the sequence of polynomials: $$…

In this paper, we prove two recently conjectured supercongruences (modulo $p^3$, where $p$ is any prime greater than $3$) of Zhi-Hong Sun on truncated sums involving the Domb numbers. Our proofs involve a number of ingredients such as…

In 1997, Van Hamme proposed 13 supercongruences on truncated hypergeometric series. Van Hamme's (B.2) supercongruence was first confirmed by Mortenson and received a WZ proof by Zudilin later. In 2012, using the WZ method again, Sun…

Let $p$ be an odd prime and $r\geq 1$. Suppose that $\alpha$ is a $p$-adic integer with $\alpha\equiv2a\pmod p$ for some $1\leq a<(p+r)/(2r+1)$. We confirm a conjecture of Sun and prove that…

Let $p>3$ be a prime, and let $m$ be an integer with $p\nmid m$. In the paper we prove some supercongruences concerning $$\align &\sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{3k}k}{54^k},\…

Let $n\geq 3$ be an integer and $p$ be a prime with $p\equiv 1\pmod{n}$. In this paper, we show that $${}_nF_{n-1}\bigg[\begin{matrix} \frac{n-1}{n}&\frac{n-1}{n}&\ldots&\frac{n-1}{n}\\ &1&\ldots&1\end{matrix}\bigg | \, 1\bigg]_{p-1}\equiv…

We prove several supercongruences involving the harmonic number of order two $H_n^{(2)}:=\sum_{k=1}^n1/k^2$. For example, if $p>5$ is prime and $\alpha$ is $p$-integral, then we can completely determine $$…

Let $p>3$ be a prime, and let $a$ be a rational p-adic integer. Let $\{B_n(x)\}$ and $\{E_n(x)\}$ denote the Bernoulli polynomials and Euler polynomials, respectively. In this paper we show that $$\sum_{k=0}^{p-1}\binom…

Let $p>3$ be a prime and let $a$ be a positive integer. We show that if $p\equiv1\pmod 4$ or $a>1$ then $$\sum_{k=0}^{\lfloor\frac34p^a\rfloor}\frac{\binom{2k}k^2}{16^k}\equiv\l(\frac{-1}{p^a}\r)\pmod{p^3}$$ with $(-)$ the Jacobi symbol,…

We establish a parametric supercongruence related to Whipple's ${}_5F_4$ formula and Dwork's dash operation. As a typical consequence, we obtain the following result: for any prime $p\equiv3\pmod4$ and odd integer $r\geq1$, $$…

Let $m>2$ and $q>0$ be integers with $m$ even or $q$ odd. We show the supercongruence $$\sum_{k=0}^{p-1}(-1)^{km}\binom{p/m-q}{k}^m\equiv0\pmod{p^3}.$$ for any prime $p>mq$. This confirms a conjecture of Sun.

Z.-W. Sun introduced three kinds of numbers: \begin{align*}S_n=\sum_{k=0}^{n}{n\choose k}^2{2k\choose k}(2k+1),\qquad s_n=\sum_{k=0}^{n}{n\choose k}^2{2k\choose k}\frac{1}{2k-1}, \end{align*} and $S_n^{+}=\sum_{k=0}^{n}{n\choose…

Through a systematic approach on generating Wilf-Zeilberger-pairs, we prove some hypergeometric identities conjectures due to Z.W. Sun, J. Guillera and Y. Zhao etc., including two Ramanujan-$1/\pi^4$, one $1/\pi^3$ formulas as well as a…

Using the following $_4F_3$ transformation formula $$ \sum_{k=0}^{n}{-x-1\choose k}^2{x\choose n-k}^2=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}^2{x+k\choose 2k}, $$ which can be proved by Zeilberger's algorithm, we confirm some special…

We prove a supercongruence modulo $p^3$ between the $p$th Fourier coefficient of a weight 6 modular form and a truncated ${}_6F_5$-hypergeometric series. Novel ingredients in the proof are the comparison of two rational approximations to…

Using the $q$-Wilf--Zeilberger method and a $q$-analogue of a "divergent" Ramanujan-type supercongruence, we give several $q$-supercongruences modulo the fourth power of a cyclotomic polynomial. One of them is a $q$-analogue of a…

In recent years, Z.-W. Sun proposed several sophisticated conjectures on congruences for finite sums with terms involving combinatorial sequences such as central trinomial coefficients, Domb numbers and Franel numbers. These sums are double…

In this paper, we confirm some congruences conjectured by V.J.W. Guo and M.J. Schlosser recently. For example, we show that for primes $p>3$, $$…

In this paper, we prove two conjectural supercongruences on the $(p-1)$th Ap\'ery number, which were recently proposed by Z.-H. Sun.